n?u=RePEc:bdi:wptemi:td_971_14&r=ets

Temi di Discussione

(Working Papers)

Random switching exponential smoothing

and inventory forecasting

by Giacomo Sbrana and Andrea Silvestrini

September 2014

Number

**971**

Temi di discussione

(Working papers)

Random switching exponential smoothing

and inventory forecasting

by Giacomo Sbrana and Andrea Silvestrini

Number **971** - September 2014

The purpose of the Temi di discussione series is to promote the circulation of working

papers prepared within the Bank of Italy or presented in Bank seminars by outside

economists with the aim of stimulating comments and suggestions.

The views expressed in the articles are those of the authors and do not involve the

responsibility of the Bank.

Editorial Board: Giuseppe Ferrero, Pietro Tommasino, Piergiorgio Alessandri,

Margherita Bottero, Lorenzo Burlon, Giuseppe Cappelletti, Stefano Federico,

Francesco Manaresi, Elisabetta Olivieri, Roberto Piazza, Martino Tasso.

Editorial Assistants: Roberto Marano, Nicoletta Olivanti.

ISSN 1594-7939 (print)

ISSN 2281-3950 (online)

Printed by the Printing and Publishing Division of the Banca d’Italia

RANDOM SWITCHING EXPONENTIAL SMOOTHING

AND INVENTORY FORECASTING*

by Giacomo Sbrana§ and Andrea Silvestrini§§

Abstract

Exponential smoothing models are an important prediction tool in macroeconomics,

finance and business. This paper presents the analytical forecasting properties of the random

coefficient exponential smoothing model in the multiple source of error framework. The

random coefficient state-space representation allows for switching between simple

exponential smoothing and the local linear trend. Therefore it is possible to control, in a

flexible manner, the random changing dynamic behaviour of the time series. The paper

establishes the algebraic mapping between the state-space parameters and the implied

reduced form ARIMA parameters. In addition, it shows that parametric mapping surmounts

the difficulties that are likely to emerge in a direct estimatation of the random coefficient

state-space model. Finally, it presents an empirical application comparing the forecast

accuracy of the suggested model vis-à-vis other benchmark models, both in the ARIMA and

in the Exponential Smoothing class. Using time series relative to wholesalers’ inventories in

the USA, the out-of-sample results show that the reduced form of the random coefficient

exponential smoothing model tends to be superior to its competitors.

JEL Classification: C22, C53.

Keywords: exponential smoothing, ARIMA, inventory, forecasting.

Contents

1. Introduction .......................................................................................................................... 5

2. The random coefficient state-space model .......................................................................... 8

2.1 The multiple source of error framework ........................................................................ 8

2.2 The mapping between the structural and the ARMA parameters .................................. 9

2.3 ARMA estimation of the random coefficient state-space model ................................. 12

3. Empirical application ......................................................................................................... 16

4. Conclusions ........................................................................................................................ 22

References .............................................................................................................................. 24

Appendix ................................................................................................................................ 27

Tables and figures................................................................................................................... 30

_______________________________________

* This is the working paper version of the article “Random switching exponential smoothing and inventory

forecasting”, forthcoming in the International Journal of Production Economics.

§ NEOMA Business School, 1 Rue du Maréchal Juin, 76130 Mont Saint Aignan, France.

§§ Bank of Italy, Directorate General for Economics, Statistics and Research.

1 Introduction 1

Exponential smoothing models represent an important prediction tool both in macroeconomics,

finance and business.

In macroeconomics, exponential smoothing is popular and widely employed due to

its simplicity and ability to capture nonstationarity. This type of process has a long tradition

and it is one of the most recommended method in forecasting time series. The

optimal properties of the simple exponential smoothing were first discussed in Muth

(1960). The exponential smoothing became the cornerstone process at the origin of

the rational expectations theory as in Muth (1961). Since then, several authors decided

to employ this model. For example, Nelson and Schwert (1977) claimed that simple

exponential smoothing is the best forecasting process, outperforming other univariate

processes, in modeling the inflation rate. Similar results have been achieved more recently

by Stock and Watson (2007).

Exponential smoothing is not only employed in macro, but also in business and finance.

In finance, this model is widely used to produce forecasts of volatilities of financial

data by the RiskMetrics methodology. This well known technique was developed

by J.P. Morgan to perform volatility prediction and is extremely popular nowadays. See

Sbrana and Silvestrini (2013) for a recent application to risk evaluation.

In business analysis, exponential smoothing models are widely employed in supply

chain management and forecasting given their simplicity, robustness and accuracy

(see Gardner, 2006; Dekker et al., 2004; Fliedner and Lawrence, 1995; Fliedner, 1999;

Moon et al., 2012, 2013; Widiarta et al., 2009). This research field has gained increased

interest because the adoption of an effective forecasting approach has important consequences

not only on the production process itself but also on the proper functioning of

the entire supply chain.

In general, based on the dynamic properties of the time series, a relevant issue often

faced by researchers and practitioners is selecting the specific exponential smoothing

1 While assuming the scientific responsibility for any errors in the paper, the authors wish to thank

Giuseppe Grande, Lyudmila Grigoryeva and Juan-Pablo Ortega for useful suggestions and discussion.

The paper is the responsibility of its authors and the opinions expressed here do not necessarily reflect

those of the Bank of Italy or the Eurosystem.

5

model. For instance, the choice between adopting a local linear trend (trend exponential

smoothing) and a simple exponential smoothing is usually driven by the detection (or

by the absence) of a trend in the data. Nevertheless, the trend dynamics of a series is not

necessarily constant over time, and may vary over the business cycle.

To deal with this issue, the time series literature has suggested some state-space

alternatives, such as the damped trend exponential smoothing, whose main feature consists

of adding an autoregressive (damping) parameter into the state-space (see Gardner

and McKenzie, 1988, 1989 and 2011). Damped trend exponential smoothing models

have gained importance in empirical studies due to their remarkable forecasting properties.

This is confirmed by Armstrong (2006), who recommends these models for

reducing forecasting error. Likewise, Li, Disney and Gaalman (2014) demonstrate the

importance of the damped trend forecasting in the supply chain, inventory and operations

management fields.

Recently, McKenzie and Gardner (2010) have proposed a random coefficient statespace

model, which is akin to the damped trend exponential smoothing. This model

introduces further flexibility as it allows for a stochastic mixture of standard linear trend

and simple exponential smoothing. However, the authors consider a single noise driving

both the trend and the slope of the state-space formulation. This is an assumption which

is relaxed hereafter by adopting a more general unobserved components framework (see

the discussion in Harvey, 2006). More specifically, this paper provides the analytical

forecasting properties of the random coefficient state-space model in the more general

multiple source of error context. In this framework, it establishes the algebraic mapping

between the state-space parameters and the implied random coefficient ARIMA parameters

(and vice versa). 2 Relying on this parametric mapping, it is proposed a procedure

that simply requires estimating an ARIMA(1,1,2) model and then deriving analytically

the random coefficient state-space parameters, keeping the optimal properties (consistency

and asymptotic normality) of the maximum likelihood estimator for the ARIMA

models. This represents a practical improvement due to the difficulties that are likely to

emerge in estimating directly the random coefficient state-space model using standard

2 The idea of expressing linear exponential smoothing recursions in terms of their reduced ARIMA

form dates back to Box and Jenkins (1976).

6

maximum likelihood techniques.

Another feature of interest is the out-of-sample predicting performance of the random

coefficient ARIMA reduced form model, which is evaluated by conducting an

extensive forecast experiment focusing on disaggregate monthly inventory time series

of merchant wholesalers in the U.S.A. It should be stressed that the out-of-sample evaluation

period includes the global financial crisis.

The interest in US inventory forecasting stems from the fact that inventory movements

at aggregate level play a key role in business cycle fluctuations (Blinder and

Maccini, 1991a, 1991b). In fact, despite making up only a relatively small fraction of

fixed investment and GDP, inventories significantly contribute to the volatility of real

GDP growth; in addition, Figure B.1 shows the positive association between industrial

production and inventory investment in the U.S.A. over the past two decades (on this

latter issue, the interested reader is referred to Wen, 2005).

The empirical results of the forecast experiment seem promising since the out-ofsample

forecasts produced by the random coefficient ARIMA are more accurate (in

terms of Mean Squared Forecast Error) than those of the competitors, both at short and

at longer lead times.

The remainder of the paper proceeds as follows. The main results are contained

in Section 2, which provides the algebraic relations between the state-space and the

implied reduced form ARIMA parameters. Section 3 presents an empirical application

dealing with inventory forecasting in the U.S.A., in which the relative performance of

the random coefficient ARIMA model is evaluated with respect to relevant benchmark

models belonging to the ARIMA class and to the Exponential Smoothing single source

of error family. Section 4 concludes.

7

2 The random coefficient state-space model

2.1 The multiple source of error framework

Consider the following random coefficient state-space model with additive noises (errors):

y t = l t−1 + A t b t−1 + ɛ t

l t = l t−1 + A t b t−1 + η t

b t = A t b t−1 + ξ t (1)

Where {y t } is the observed time series, {l t } is its level (or stochastic trend), {b t } is

the slope of its stochastic trend, and t = 1, 2, . . . , N is the number of observations.

This is a standard state-space representation except that, in all the equations, the term

A t represents a sequence of independent, identically distributed binary random variates

with probability P (A t = 1) = φ and probability P (A t = 0) = (1 − φ).

It is assumed that the noises have zero mean and are uncorrelated, that is:

⎛ ⎞ ⎛

⎞

ɛ t σɛ 2 0 0

cov ⎜

⎝ η t

⎟

⎠ = ⎜

⎝ 0 ση 2 0 ⎟

⎠ (2)

ξ t 0 0 σξ

2

These variances are also defined structural parameters (see Harvey, 1989).

Note that equations (1) and (2) represent the generalization of the random coefficient

state-space models as in McKenzie and Gardner (2010) to the multiple source

of error framework (given the three uncorrelated noises characterising the dynamics of

y t ). 3 When A t = φ this state-space representation collapses to the damped trend model

in the multiple source of error form.

The random coefficient state-space model is particularly flexible since it allows for a

mixture of local linear trend and simple exponential smoothing models. This feature is

not only appealing from a modeling viewpoint, but it is also in line with Brown (1963),

3 The hypothesis of uncorrelated errors can be relaxed. However, for the sake of simplicity of the

algebraic calculations we keep the noises uncorrelated.

8

who argued that the parameters of the model may change from one segment to another

as processes are thought to be locally constant.

To better understand this point and its relevance in empirical analysis, Figure B.2

shows a simulated time series for the model (1) and (2), with 160 observations. The

series is generated such that every 20 observations the model switches from a simple

exponential smoothing (that is A t = 0 for the intervals: 0-20; 40-60; 80-100; 120-140)

to a local linear trend (that is A t = 1 for the intervals: 20-40; 60-80; 100-120; 140-160).

Indeed, the changing behavior of the series represents a relevant feature since it adapts

to the switching dynamics of many economic and financial time series.

In what follows some algebraic steps are needed before showing how the structural

parameters determine the forecasting properties of the model in its stationary form.

More specifically, these algebraic steps are instrumental for providing the exact link

between the parameters of the state-space model in (1) and (2) and those of its stationary

ARMA form.

2.2 The mapping between the structural and the ARMA parameters

The stationary representation of the process (1) with (2) can be derived by taking the

first differences, such that:

z t = y t −y t−1 = l t−1 −l t−2 +A t b t−1 −A t−1 b t−2 +ɛ t −ɛ t−1 = ɛ t −ɛ t−1 +η t−1 +A t b t−1

Moreover:

(1 − A t L)z t = ɛ t − ɛ t−1 + η t−1 + A t b t−1 − A t ɛ t−1 + A t ɛ t−2 − A t η t−2 − A t A t−1 b t−2 =

= ɛ t − ɛ t−1 − A t ɛ t−1 + A t ɛ t−2 + η t−1 − A t η t−2 + A t ξ t−1 (3)

where L is the lag operator (i.e., Lz t = z t−1 ). The last expression holds since A t b t−1 −

A t A t−1 b t−2 = A t ξ t−1 . Therefore equation (3) is a random coefficient ARIMA(1,1,2)

model, that is a mixture of two ARIMA models. More specifically, an ARIMA(0,2,2)

(1 − L)z t = ɛ t − 2ɛ t−1 + ɛ t−2 + η t−1 − η t−2 + ξ t−1

9

with probability φ and an ARIMA(0,1,1)

z t = ɛ t − ɛ t−1 + η t−1

with probability (1 − φ).

The following Proposition further clarifies the dynamic properties of the stationary

process z t .

PROPOSITION 1. Given (1) and (2), the autocovariances of z t are:

∞

E(z t z t ) = 2σɛ 2 + σ2 η + ∑

φ i σξ 2 = 2σ2 ɛ + σ2 η + φ

(1 − φ) σ2 ξ

i=1

E(z t z t−1 ) = −σ 2 ɛ + φ2

(1 − φ) σ2 ξ

E(z t z t−2 ) =

φ 3

(1 − φ) σ2 ξ

E(z t z t−n ) = φn+1

(1 − φ) σ2 ξ = φn−2 E(z t z t−2 ) ∀n ≥ 3 (4)

Proof. The proof can be found in the Appendix.

Therefore, the resulting process for the variable y t is an ARIMA(1,1,2) with autoregressive

parameter equal to φ since its autocorrelation functions are ρ(n) = φ n−2 ρ(2)

for any n ≥ 2. Hence the z t process can be written as:

(1 − φL)z t = a t + θ 1 a t−1 + θ 2 a t−2 (5)

Note that the variance of a t , i.e., σa 2 , represents the theoretical forecast error variance

of the random coefficient state-space model in (1) and (2). The same reduced form (5)

holds in the single source of error case as shown by McKenzie and Gardner (2010).

Before showing the mapping of the random coefficient state-space parameters to the

ARIMA parameters, it is relevant to derive the autocovariances of the right-hand side of

(5).

10

One can see that, given (1) with (2), the autocovariances of the right-hand side of

(5) are:

γ 0 = E[(z t − φz t−1 ) 2 ] = (1 + φ 2 )E(z t z t ) − 2φE(z t z t−1 ) =

= ση 2 + φ(σξ 2 + (ση 2 + σξ)φ) 2 + 2σɛ 2 (1 + φ + φ 2 )

γ 1 = E[(z t − φz t−1 )(z t−1 − φz t−2 )] =

= (1 + φ 2 )E(z t z t−1 ) − φE(z t z t−2 ) − φE(z t−1 z t−1 ) = −σ 2 ηφ − σ 2 ɛ (1 + φ) 2

γ 2 = E[(z t − φz t−1 )(z t−2 − φz t−3 )] = E(z t z t−2 ) − φE(z t z t−1 ) = φσ 2 ɛ

γ n = E((z t − φz t−1 )(z t−n − φz t−n−1 )) =

= (1 + φ 2 )E(z t z t−n ) − φE(z t z t−n−1 ) − φE(z t z t−n+1 ) =

= (1 + φ 2 )φ n−2 E(z t z t−2 ) − φ n−2 E(z t z t−2 ) − φ n E(z t z t−2 ) = 0 ∀n ≥ 3

(6)

We are now able to map the structural parameters as in (1) and (2) into the reduced

form parameters in (5). This is the content of Proposition 2.

PROPOSITION 2. Given (1) and (2), the reduced form parameters of (5) are analytical

functions of the structural parameters as in what follows:

σa 2 = 1 4 (2σ2 ɛ + σ2 η + σ2 ξ φ + (2σ2 ɛ + σ2 η + σ2 ξ )φ2 + D √ 1 + φ+

+ √ √

2 −2σɛ 4 − 12σɛ 4 φ 2 − 8σɛ 2 σηφ 2 2 − 2σηφ 4 2 + 4σɛ 2 σξ 2φ2 + 4σɛ 2 σξ 2φ3 − 2σɛ 4 φ 4 + F 2 + F D √ 1 + φ )

θ 2 = φσ2 ɛ

σ 2 a

θ 1 = −σ2 ηφ−σ 2 ɛ (1+φ) 2

φσ 2 ɛ +σ2 a

(7)

with

11

√

D =

√ση 2 + (σ2 η + σ2 ξ )φ + 4σ2 ɛ (1 + φ) ση 2(−1 + φ)2 + σξ 2 φ(1 + φ)

F = 2σ 2 ɛ + σ2 η + σ2 ξ φ + (2σ2 ɛ + σ2 η + σ2 ξ )φ2

Proof. The proof can be found in the Appendix. 4

Remark 1. Note that the suggested framework as in (1) and (2) implies the following

conditions on the reduced form ARIMA model: θ 2 > 0 and θ 1 < 0. Indeed, the

expressions as in (6) guarantee that γ 1 < 0 and γ 2 > 0.

Remark 2. The algebraic results are valid in general regardless of the distribution of

the noises of the model (1) and (2).

Figures B.3 and B.4 display three-dimensional and contour plots of the variance σ 2 a

as a function of γ 1 and γ 2 . In particular, Figure B.3 shows the variance as a function of

γ 1 and γ 2 , fixing γ 0 = 1. Figure B.4 refers to the corresponding contour plot.

These expressions give full control of the theoretical variance of the random coefficient

state-space model. 5 As a consequence, the minimum mean squared error (MMSE)

forecasts as well as the h-steps ahead theoretical forecast error variance are also algebraic

functions of the structural parameters.

Figures B.5 and B.7 display three-dimensional plots of θ 1 and θ 2 as a function of γ 1

and γ 2 (fixing γ 0 = 1). Figures B.6 and B.8 show the corresponding contour plots. In

the graphs, it is important to note that the regions of the ARIMA parameters spanned by

the model (1) and (2) are in accordance with the constraints described in Remark 1.

2.3 ARMA estimation of the random coefficient state-space model

The direct estimation of model (1) with (2) is a challenging task. Here we present a

simple estimator that enjoys the same asymptotic properties of the maximum likelihood

estimator for ARMA processes. It is relevant to note that the number of structural

4 These analytical expressions are derived using Mathematica. A file containing all calculations is

available upon request.

5 An Eviews 7 program containing a Monte Carlo simulation running the whole procedure can be

provided by the authors upon request. In addition, an Excel file computing the autocovariances as in (6)

as well as the moving average parameters, given the structural parameters, is also available upon request.

12

γ 2 = σ 2 aθ 2

(9)

parameters coincides with that of the reduced form. Indeed, there are four structural parameters

(φ; σɛ 2; σ2 η ; σ2 ξ ) mapping four ARIMA parameters (φ; θ 1; θ 2 ; σa 2 ). Therefore the

mapping given in Proposition 2 can be reversed as shown in the following Proposition.

PROPOSITION 3. Given (1) and (2), the structural parameters of the random coefficient

state-space model can be derived as analytical functions of the reduced form

parameters such that:

⎡

Λ =

⎢

⎣

φ

σξ

2

ση

2

σɛ

2

⎤ ⎡

=

⎥ ⎢

⎦ ⎣

φ

σ 2 a(θ 2 +φ(θ 1 +φ))(1+φ(θ 1 +θ 2 φ))

φ 3 (1+φ)

− σ2 a(θ 1 φ+θ 2 (1+φ(2+θ 1 +φ)))

φ 2

σ 2 aθ 2

φ

⎤

⎥

⎦

(8)

Proof. The expression for φ follows from the proof of Proposition 1 (see the Appendix).

The remaining parameters are derived by solving the system of three equations (the

autocovariances of the right-hand side of (5)) with respect to the three variances (i.e.,

σɛ 2 ; σξ 2; σ2 η). That is, by solving the following system of equations:

γ 0 = σa(1 2 + θ1 2 + θ2)

2

γ 1 = σa 2 (θ 1 + θ 1 θ 2 )

Where the γ’s are given in (6).

Remark 3. It is relevant to note that the algebraic link between the parameters of the

process as in (1) and those of the ARIMA(1,1,2) as in (5) is guaranteed if and only

if the derived parameters in the left-hand side of (8) are all positive (this implies that

γ 1 < 0 and γ 2 > 0 as noted in Remark 1). Indeed, these parameters must be positive

13

y definition (see (2)). Moreover, this condition enables the indirect estimation of the

parameters as in (1).

An important consequence of the closed-form results as in Proposition 3 is the following

limiting result.

THEOREM 1. Given (1), (2) and (8). Let:

⎡ ⎤

φ

θ 1

β =

θ 2

⎢

⎣ σa

2 ⎥

⎦

then:

ˆΛ → Λ

where → denotes convergence in probability.

In addition, if also the fourth moment of a t is finite, then the following holds:

√ )

N

(ˆΛ − Λ ⇒ N(0,

( ) T ∂Λ

Σ

∂β

( ) ∂Λ

) (10)

∂β

where ⇒ denotes convergence in distribution to the Gaussian and T denotes the transpose.

In addition:

14

⎡

Σ =

⎢

⎣

c 1 c 2 c 3 0

c 2 c 4 c 5 0

c 3 c 5 c 6 0

0 0 0 2σa

4

⎤

⎥

⎦

with 6 c 1 = − (−1+φ2 )(1+φ(θ 1 +θ 2 φ)) 2

(θ 2 +φ(θ 1 +φ)) 2

c 2 = − (−1+θ 2)(1+θ 2 +θ 1 φ)(−1+φ 2 ) (1+φ(θ 1 +θ 2 φ))

(θ 2 +φ(θ 1 +φ)) 2

c 3 = − (−1+θ 2)(θ 1 +φ+θ 2 φ)(−1+φ 2 ) (1+φ(θ 1 +θ 2 φ))

(θ 2 +φ(θ 1 +φ)) 2

c 4 = − (−1+θ 2)(1+θ 2 +2θ 1 (1+θ 2 )φ+(−1+2θ 2 1 +θ 2+3θ 2 2 +θ3 2 )φ2 +2θ 1 θ 2 (1+θ 2 )φ 3 +(1+θ 2 1 (−1+θ 2)+θ 2 )φ 4 )

(θ 2 +φ(θ 1 +φ)) 2

c 5 = − (−1+θ 2)(θ 3 1 φ2 +(−1+θ 2 )(1+θ 2 ) 2 φ(−1+φ 2 )+θ 1 (1+θ 2 φ 2 ) 2 +θ 2 1 (1+θ 2)(φ+φ 3 ))

(θ 2 +φ(θ 1 +φ)) 2

c 6 = − (−1+θ 2)(2θ 1 (1+θ 2 )φ(1+θ 2 φ 2 )+θ 2 1 (1+θ 2(−1+2φ 2 ))+(1+θ 2 )(φ 2 +θ 2 (θ 2 −(−2+θ 2 )φ 2 +θ 2 φ 4 )))

(θ 2 +φ(θ 1 +φ)) 2

and

( ) ∂Λ

=

∂β

⎡

⎢

⎣

1 c

0

0

⎤

σa(θ 2 1 φ+θ 2 (2+(2+θ 1 )φ))

− σ2 aθ 2

φ 3 φ 2

− σ2 a(1+φ(2+θ 1 +φ)) σ 2

φ 2

a

φ

− θ 1φ+θ 2 (1+φ(2+θ 1 +φ)) θ 2φ ⎥

φ 2 ⎦

σ 2 a(1+θ 2 +2θ 1 φ+(1+θ 2 )φ 2 )

φ 2 (1+φ)

− σ2 a(1+θ 2 )

φ

0

σ 2 a(1+2θ 2 φ 2 +φ 4 +θ 1 (φ+φ 3 ))

φ 3 (1+φ)

0 (θ 2+φ(θ 1 +φ))(1+φ(θ 1 +θ 2 φ))

φ 3 (1+φ)

6 The derivation of the matrix Σ was obtained with Time Series 1.4.1 package of Mathematica.

15

with

c = σ2 a(−3θ 2 −2(θ 1 +(2+θ 1 )θ 2 )φ−(1+θ 2 1 +θ2 2 +3θ 1(1+θ 2 ))φ 2 −2(1+θ 2 1 +θ2 2 )φ3 −(θ 1 +(−1+θ 1 )θ 2 )φ 4 )

φ 4 (1+φ 2 )

Proof. The convergence in probability of the maximum likelihood estimator for ARMA

processes is shown in Brockwell and Davis (1991) (see Chapter 10, Theorem 10.8.1).

Therefore this property also holds for Λ, due to the continuous mapping theorem. The

asymptotic normality is also shown in Chapter 10, Brockwell and Davis (1991) (see

Theorem 10.8.2). This also holds for Λ due to the Delta method, given that Λ is a simple

function of β. Note that, in (10), Σ represents the asymptotic variance of the maximum

likelihood estimator for ARMA processes (derived in Box and Jenkins, 1976, Chapter

7). While ∂Λ represents the partial derivative of Λ with respect to the vector of ARMA

∂β

parameters β.

These results have important practical consequences. Indeed, one can easily estimate

an ARIMA(1,1,2) model and then derive the structural parameters still relying on

the optimal properties of the maximum likelihood estimator for the ARIMA processes.

In the next section an empirical application using these analytical results is presented.

3 Empirical application

This section focuses on an empirical application dealing with inventory forecasting.

It first describes the data and the experimental set-up; then, it discusses the forecasting

results. The aim is to evaluate the relative forecasting performance of the reduced

form of the random coefficient state-space model in equations (1) and (2) – which is an

ARIMA(1,1,2) with restrictions on the parameter space – with respect to competitive

models belonging to the ARIMA class and to the Exponential Smoothing single source

of error (ETS) family (Hyndman et al., 2008).

The variables used for the purpose of our research are seasonally adjusted estimates

of monthly sector-level inventories of merchant wholesalers (except manufacturers’

sales branches and offices), expressed in millions of dollars, at different aggregation

level. Data refer to the 4-digit North American Industry Classification System (NAICS)

industry level durable (nine time series) and non-durable goods (nine time series), to

16

the 3-digit aggregate durable and non-durable series (NAICS 423 – total durable wholesale

and NAICS 424 – total non-durable wholesale) and to the aggregate 2-digit total

merchant wholesalers time series (NAICS 42, total wholesale). The source of the data

is U.S. Census Bureau’s (http://www.census.gov/wholesale/). 7 For each

series, 261 observations ranging from January 1992 through September 2013 are available.

A full description of the dataset is provided in Table 1.

The (4-digit and 3-digit) time series plots are presented in Figure B.9. In the vertical

axis, the outstanding amounts are expressed in thousands of millions of dollars.

In order to evaluate the forecasting performance of the random coefficient ARIMA

model (RC-ARIMA(1,1,2)), the exercise begins by considering a group of three benchmark

ARIMA models:

1. ARIMA(p,1,q) model, with p = 1, 2, 3; q = 1, 2, 3 (hence, the maximum AR and

MA order is set equal to three); the standard information criteria are applied in

order to select p and q (in particular, the Schwartz Information Criterion, BIC);

2. ARIMA(0,1,1) model, which is the reduced form of simple exponential smoothing;

3. ARIMA(0,2,2) model, which is the reduced form of a local linear trend model

(see, e.g., Harvey, 1989).

All the models are specified (and estimated) with a constant. The data are in levels.

The set-up of the forecasting exercise is as follows. For each time series, the first

in-sample used for estimation ranges from 1992/01 to 2008/12. The out-of-sample evaluation

period goes from 2009/01 onwards. Consistently, in Figure B.9 the shaded areas

identify the out-of-sample, which fully includes the global financial crisis. This makes

the whole forecasting exercise more challenging.

The maximum forecast horizon is set to six months. Thus, h = 1, 3, 6-steps-ahead

predictions are made for 2009/01, 2009/03, and 2009/06, using the RC-ARIMA(1,1,2)

and the three benchmark ARIMA models as above.The out-of-sample forecasting accuracy

of the RC-ARIMA(1,1,2) model is evaluated by its Mean Squared Forecast Error

(MSFE) compared to that of a competitive model.

7 Accessed on December 4, 2013.

17

For each prediction model, forecast horizon and information set, the MSFE is defined

as:

MSF E N t 0

(h, j) =

1

N − t 0 + 1

N∑

t=t 0

(y t+h − ŷ j t+h|t )2 (11)

where y t+h is the realized value of the variable at time t + h, t 0 and N are the first

and last data point in the out-of-sample, ŷ j t+h|t

is the forecast made at time t at horizon

h = 1, 3, 6 from model j. Relative MSFEs are calculated simply dividing the MSFE of

the RC-ARIMA(1,1,2) model by the MSFE of one of the benchmark models. Values

greater than one for the relative MSFE indicate that the MSFE produced by the random

coefficient state-space model is larger than that of the competitive model. Conversely,

ratios below unity indicate that random coefficient state-space model forecasts are more

accurate with respect to the competitive model.

Then, the in-sample is expanded by one observation, the models are re-estimated,

h = 1, 3, 6-steps-ahead forecasts are again produced (this time for 2009/02, 2009/04,

and 2009/07) and the corresponding (relative) MSFEs are calculated. The same procedure

is repeated recursively 55 times. Thereby, we get a sequence of forecast errors on

which the MSFEs are based.

Each h = 1, 3, 6-steps-ahead prediction is evaluated over 50 observations: this

means that, for instance, when considering h = 1-step-ahead predictions, the first forecast

point is 2009/01, while the last forecast point is 2013/02. Similarly, for h = 6-

steps-ahead predictions, the first forecast point is 2009/06, whilst the last forecast point

is 2013/07.

Every time the RC-ARIMA(1,1,2) model is estimated, it is carefully checked that

Remark 3 is satisfied. This guarantees that σξ 2 > 0, σ2 η > 0 and σ2 ɛ > 0. This check is repeated

each time a new observation is added to the in-sample from the out-of-sample and

forecasts are produced. If these constraints are not met by the estimated ARIMA(1,1,2)

parameters, then the corresponding time series is excluded from the competition. In

this exercise, for instance, nine time series have been discarded (S4235, S4239, S424,

S4241, S4242, S4244, S4245, S4247 and S4249).

The following tables summarise the accuracy of the models, displaying the relative

18

MSFEs and the Diebold and Mariano (1995) forecast accuracy test results. Table 2

refers to the one-step-ahead forecast results. The three and six-steps ahead projections

results are presented in Tables 3, and 4, respectively.

All the tables have the same structure. Each row corresponds to a single inventory

time series. The second column reports the ratio of MSFE for the RC-ARIMA(1,1,2) to

the MSFE for the ARIMA(p,1,q); the third column refers to the ratio of MSFE for RC-

ARIMA(1,1,2) vs ARIMA(0,1,1), while the fourth column refers to the ratio of MSFE

for RC-ARIMA(1,1,2) vs ARIMA(0,2,2). For completeness’ sake, results relative to the

nine items excluded from the forecasting competition are reported in the bottom part of

the tables.

The last three columns report the Diebold and Mariano (DM) test results. In particular,

column five compares the performance of RC-ARIMA(1,1,2) vs ARIMA(p,1,q);

column six refers to the comparison between RC-ARIMA(1,1,2) and ARIMA(0,1,1);

column seven refers to the comparison between RC-ARIMA(1,1,2) and ARIMA(0,2,2).

The basis of the DM test statistics is the sample mean of the observed loss differential

series over all the 50 forecast points. In this application, a quadratic loss function has

been used. Under the null hypothesis of equal predictive accuracy (i.e., equal expected

loss) the limiting distribution of the test statistics is standard Normal. A DM test with

associated p-value smaller than 0.05 is shown in bold, implying a rejection of the null

hypothesis (equal forecasting accuracy) at 5 percent level. Negative (positive) values

of the DM test indicate that the RC-ARIMA(1,1,2) performs better (worse) than the

alternative model.

In summary, what emerges from Tables 2 to 4 is the following:

• The most striking result is that RC-ARIMA(1,1,2) model performs clearly better

than the ARIMA(0,1,1) and ARIMA(0,2,2) benchmarks, at all forecast horizons.

The ARIMA(0,2,2) model is never successful compared to the RC-ARIMA(1,1,2).

The ARIMA(0,1,1) has better performance only for the S4248 series, when h = 6.

Therefore, substantial improvements can be achieved by using the RC-ARIMA(1,1,2)

model, which allows for switching between the simple exponential smoothing and

the local linear trend models and therefore provides with a more flexible parameterization

of the changing dynamic behavior of the time series;

19

• When the ARIMA(p,1,q) benchmark is considered, results are less clear-cut. The

RC-ARIMA(1,1,2) model is slightly superior at short horizons (h = 1, 3), whereas

at longer horizons (h = 6) the reverse is true. However, in general most of the

times the DM test statistics is not significant. This is largely expected. Indeed,

when considering the ARIMA(p,1,q) benchmark, standard model selection criteria

based on the log-likelihood function (with a penalty term for the number of

parameters in the model) such as the BIC are used to select the “best” AR and MA

order. 8 Yet, in-sample information criteria based on the penalized log-likelihood

do not always constitute the most reliable indicators for choosing the best forecasting

model;

• The ranking of the RC-ARIMA(1,1,2) relative to the ARIMA(p,1,q), ARIMA(0,1,1)

and ARIMA(0,2,2) models does not appear to depend on the forecast horizon;

• Looking at the excluded items, the DM test results suggest that most of the times

there are no major differences in predictive ability between pairs of models.

Next, the forecasting performance of the RC-ARIMA(1,1,2) is evaluated with respect

to competitive models belonging to the Exponential Smoothing single source of

error (ETS) family. Although it is well known that ARIMA models and the ETS family

overlap and are complementary (see the discussion in Hyndman et al., 2008), ETS

models have been shown to perform very well in business forecasting and are now a

benchmark in most forecasting competitions.

Specifically, three models are considered (the same taxonomy as in Hyndman et al.,

2008, is used in what follows):

1. ET S(A, A d , N): this is the damped trend as, for instance, in Gardner and McKenzie

(2011); it has an ARIMA(1,1,2) reduced form;

2. ET S(A, A, N): this is the additive Holt’s linear method, which has an ARIMA(0,2,2)

reduced form;

3. ET S(A, N, N): this is the simple exponential smoothing, which has an ARIMA(0,1,1)

reduced form.

8 Actually, this procedure appears to be employed by most econometric software packages.

20

The ET S(A, A d , N) damped trend is the most general specification. Since the pioneering

contributions of Gardner and McKenzie (1988, 1989), damped trend exponential

smoothing gained importance due to its remarkable forecasting properties. Therefore,

the damped trend is often considered as a benchmark model to beat in forecasting

competitions.

The set-up of the forecasting exercise is as before. To fit the ETS models and produce

forecasts, the ETS exponential smoothing built-in procedure provided in EViews

8 is used. 9 The results are presented in Tables from 5 to 7.

Overall, the RC-ARIMA(1,1,2) model shows good forecasting ability, performing

better than the ETS benchmarks at all forecast horizons. According to the DM test results,

the differences in the forecast error quadratic loss differential are further enhanced,

as shown by the statistical significance of these statistics.

Interestingly, looking in more detail to Tables 5 to 7 reveals that:

• In general, the RC-ARIMA(1,1,2) model is clearly overperforming the three ETS

benchmarks; this is also evident when the comparison is made with the ET S(A, A d , N);

• The ET S(A, A, N)) and ET S(A, N, N) models are rapidly outperformed by the

RC-ARIMA(1,1,2), which is unsurprising in the light of the swings observed by

most of the inventory series during the out-of-sample evaluation period. Considering

the damped trend version ET S(A, A d , N) does not seem to help in this

respect, given that the DM test results reveal that even this model is clearly outperformed

by the RC-ARIMA(1,1,2), especially at very short horizons;

• Also in this case, the ranking of the RC-ARIMA(1,1,2) with respect to its competitors

does not appear to depend on the chosen forecast horizon;

• Focusing on the excluded items, according to the DM test, no major differences

in predictive ability are detected.

9 The Average Mean Square Error is the objective function, meaning that the estimated parameter

values and initial state values minimize the Average Mean Square Error of the h-step forecasts of the

specified ETS model.

21

Summing up, this application provides some evidence that the proposed ARIMA

model is able to outperform a variety of benchmark competitors, both in the ARIMA

and in the ETS class, on a time horizon ranging from one up to six months ahead. As

expected, the accuracy of forecasts produced by the RC-ARIMA(1,1,2) model tends to

be comparable (although slightly superior) to that of ARIMA(p,1,q) models in which

the AR and MA orders are selected by an in-sample information criterion. Moreover,

the RC-ARIMA(1,1,2) seems to outperform the considered ETS models in most of the

cases.

It should be noted, nevertheless, that these encouraging results are only valid when

some specific constraints hold (see Remark 3). This condition should be checked empirically

when using data. Indeed, this has evident consequences in our empirical exercise:

when choosing an ARIMA model, if the ARIMA(1,1,2) process has (estimated) parameters

respecting Remark 3, then it might represent a strong candidate in forecasting

inventories. If, on the other hand, conditions in Remark 3 are not met, then this model

might not be a good candidate. This should not prevent from using the forecasts of the

ARIMA(1,1,2) process. However, if the ARIMA(1,1,2) does not meet Remark 3, our

results are not as remarkable as for the RC-ARIMA case.

4 Conclusions

This paper has investigated the analytical properties of the random coefficient statespace

model with multiple source of error, generalizing the results provided by McKenzie

and Gardner (2010), who introduced this model as a variant of the damped trend

exponential smoothing in the single source of error class.

The random coefficient state-space model allows for a stochastic mixture of simple

exponential smoothing and local linear trend time series models. It is akin to the

damped trend exponential smoothing, in which an autoregressive-damping parameter is

introduced to damp the trend component. Yet, differently from the damped trend exponential

smoothing, the autoregressive-damping parameter is not constant but is instead

a random coefficient governing the persistence of the trend, which is allowed to change

suddenly. Indeed, it is widely recognised that trend persistence is not a stable time se-

22

ies property, rather it may change over the phases of the business cycle, even abruptly.

Accordingly, this specification accommodates unsmooth changes of the gradient of the

trend, adapting very well to the switching behavior of many sales and inventories time

series (especially during crisis periods).

One contribution of this paper is providing the algebraic mapping between the statespace

parameters and the implied ARIMA parameters. Relying on this parametric mapping,

it is proposed a procedure that requires simply to estimate an ARIMA(1,1,2) model

and then to derive the random coefficient state-space parameters, keeping the optimal

properties (consistency and asymptotic normality) of the maximum likelihood estimator

for the ARIMA models. This is an improvement to the literature, due to the difficulties

that are likely to emerge in estimating directly the random coefficient state-space model.

Yet, the indirect estimation of the random coefficient state-space parameters is feasible if

and only if Remark 3 is met. This represents a guideline for modelers: when the expressions

as in (8) are all positive the ARIMA(1,1,2) can be considered as reparametrization

of the random switching whose parameters can be easily derived. When the reverse is

true, the link with the ARIMA model is not legitimate; this represents a signal that the

data may not follow the stochastic dynamics as described in (1). These criteria should

be employed by practitioners in empirical analysis.

Another feature of interest is the out-of-sample predicting performance of the RC-

ARIMA reduced form model, which is evaluated by conducting an extensive forecast

experiment focusing on disaggregate monthly inventory time series of merchant wholesalers

in the U.S.A. The out-of-sample evaluation period includes the global financial

crisis. The results are encouraging since the forecasts produced by the ARIMA model

are overall more accurate than those of several relevant benchmark models belonging

both to the ARIMA class and to the Exponential Smoothing single source of error family.

These results hold true both at short and at longer lead times.

Admittedly, further research is needed to explore the forecasting capabilities of the

RC-ARIMA(1,1,2) model, which, however, deems in our view to be considered as a

useful prediction tool in macroeconomics, finance and business.

23

References

[1] Armstrong, J. S. (2006). Findings from evidence-based forecasting: Methods for

reducing forecast error. International Journal of Forecasting, vol. 22, pp. 583–598.

[2] Blinder, A. S., & Maccini, L. J. (1991a). The resurgence of inventory research:

What have we learned Journal of Economic Surveys, vol. 5, pp. 291–328.

[3] Blinder, A. S., & Maccini, L. J. (1991b). Taking stock: A critical assessment of

recent research on inventories. Journal of Economic Perspectives, vol. 5, pp. 73–

96.

[4] Box, G. E. P., & Jenkins, G. M. (1976). Time Series Analysis: Forecasting and

Control. San Francisco: Holden-Day.

[5] Brockwell, P. J., & Davis, R.A. (1991). Time Series: Theory and Methods. New

York: Springer-Verlag.

[6] Brown, R. G. (1963). Smoothing, Forecasting and Prediction of Discrete Time

Series. Englewood Cliffs, NJ: Prentice-Hall, Inc.

[7] Dekker, M., van Donselaar, K. H., & Ouwehand, P. (2004). How to use aggregation

and combined forecasting to improve seasonal demand forecasts. International

Journal of Production Economics, vol. 90, pp. 151–167.

[8] Diebold, F. X., & Mariano, R. S. (1995). Comparing predictive accuracy. Journal

of Business and Economic Statistics, vol. 13, pp. 253–263.

[9] Fliedner, G. (1999). An investigation of aggregate variable time series forecast

strategies with specific subaggregate time series statistical correlation. Computers

and Operations Research, vol. 26, pp. 1133–1149.

[10] Fliedner, E. B., & Lawrence, B. (1995). Forecasting system parent group formation:

an empirical application of cluster analysis. Journal of Operations Management,

vol. 12, pp. 119–130.

24

[11] Gardner, E. S., Jr. (2006). Exponential smoothing: The state of art-Part II. International

Journal of Forecasting, vol. 22, pp. 637–666.

[12] Gardner, E. S., Jr. & McKenzie, E. (1988). Model identification in exponential

smoothing. Journal of the Operational Research Society, vol. 39, pp. 863–867.

[13] Gardner, E. S., Jr., & McKenzie, E. (1989). Seasonal exponential smoothing with

damped trends. Management Science, vol. 35, pp. 372–376.

[14] Gardner, E. S., Jr., & McKenzie, E. (2011). Why the damped trend works. Journal

of the Operational Research Society, vol. 62, pp. 1177–1180.

[15] Harvey, A. C. (1989). Forecasting Structural Time Series and the Kalman Filter.

Cambridge: Cambridge University Press.

[16] Harvey, A. C. (2006). Forecasting with Unobserved Components Time Series

Models. In: G. Elliott, C.W.J. Granger and A. Timmermann (Eds.), Handbook

of Economic Forecasting. Vol. 1, number 1. Elsevier, Amsterdam, pp. 327–412.

[17] Hyndman, R. J., Koehler, A. B., Ord, J. K., & Snyder, R. D. (2008) Forecasting

with Exponential Smoothing: The State Space Approach. Berlin: Springer Verlag.

[18] Li, Q., Disney, S. M., & Gaalman, G. (2014). Avoiding the bullwhip effect using

Damped Trend forecasting and the Order-Up-To replenishment policy. International

Journal of Production Economics, vol. 149, pp. 3–16.

[19] McKenzie, E., & Gardner, E. S., Jr. (2010). Damped trend exponential smoothing:

a modeling viewpoint. International Journal of Forecasting, vol. 26, pp. 661–665.

[20] Moon, S., Hicks, C., & Simpson, A., (2012). The development of a hierarchical

forecasting method for predicting spare parts demand in the South Korean navy – a

case study. International Journal of Production Economics, vol. 140, pp. 794–802.

[21] Moon, S., Simpson, A., & Hicks, C., (2013). The development of a classification

model for predicting the performance of forecasting methods for naval spare parts

demand. International Journal of Production Economics, vol. 143, pp. 449–454.

25

[22] Muth, J. F. (1960). Optimal properties of exponentially weighted forecasts. Journal

of the American Statistical Association, vol. 55, pp. 299–306.

[23] Muth, J. F. (1961). Rational expectations and the theory of price movements.

Econometrica, vol. 29, pp. 315–335.

[24] Nelson, C. R. & Schwert, G. W. (1977). Short-term interest rates as predictors

of inflation: On testing the hypothesis that the real rate of interest is constant.

American Economic Review, vol. 67, pp. 478–486.

[25] Sbrana, G. & Silvestrini, A. (2013). Aggregation of exponential smoothing processes

with an application to portfolio risk evaluation. Journal of Banking and

Finance, vol. 37, pp. 1437–1450.

[26] Stock, J. H. & Watson, M. W. (2007). Why has inflation become harder to forecast

Journal of Money, Credit, and Banking, vol. 39, pp. 3–34.

[27] Wen, Y., (2005). Understanding the inventory cycle. Journal of Monetary Economics,

vol. 52, pp. 1533–1555.

[28] Widiarta, H., Viswanathan, S., & Piplani, R. (2009). Forecasting aggregate demand:

an analytical evaluation of top-down versus bottom-up forecasting in a production

planning framework. International Journal of Production Economics, vol.

118, pp. 87–94.

26

APPENDIX

A Proof of Proposition 1

It is relevant to expand equation (3) in order to derive the recursion for z t . That is,

how z t is generated by the three noises of the model (1) and by the random coefficient

parameter A t :

z t = ɛ t − (1 + A t )ɛ t−1 + A t ɛ t−2 + η t−1 − A t η t−2 + A t ξ t−1 +

+A t (ɛ t−1 − (1 + A t−1 )ɛ t−2 + A t−1 ɛ t−3 + η t−2 − A t−1 η t−3 + A t−1 ξ t−2 + A t−1 z t−2 ) =

= ɛ t − ɛ t−1 − A t A t−1 ɛ t−2 + A t A t−1 ɛ t−3 + η t−1 − A t A t−1 η t−3 + A t ξ t−1 +

+A t A t−1 ξ t−2 + A t A t−1 z t−2

In addition, we have that:

A t A t−1 z t−2 = A t A t−1 ɛ t−2 − A t A t−1 ɛ t−3 − A t A t−1 A t−2 A t−3 ɛ t−4 +

+A t A t−1 A t−2 A t−3 ɛ t−5 + A t A t−1 η t−3 − A t A t−1 A t−2 A t−3 η t−5 +

+A t A t−1 A t−2 ξ t−3 + A t A t−1 A t−2 A t−3 ξ t−4 + A t A t−1 A t−2 A t−3 z t−4

such that the expression for z t reduces to:

z t = ɛ t − ɛ t−1 − A t A t−1 A t−2 A t−3 ɛ t−4 + A t A t−1 A t−2 A t−3 ɛ t−5 +

+η t−1 − A t A t−1 A t−2 A t−3 η t−5 + A t ξ t−1 + A t A t−1 ξ t−2 + A t A t−1 A t−2 ξ t−3 +

+A t A t−1 A t−2 A t−3 ξ t−4 + A t A t−1 A t−2 A t−3 z t−4

The generic recursion for z t is as follows:

z t = ɛ t − ɛ t−1 − A t A t−1 · · · A t−(2n−1) ɛ t−2n + A t A t−1 · · · A t−(2n−1) ɛ t−(2n+1)

+η t−1 − A t A t−1 · · · A t−(2n−1) η t−(2n+1) + A t ξ t−1 + A t A t−1 ξ t−2 +

+A t A t−1 A t−2 ξ t−3 + · · · + A t A t−1 · · · A t−(2n−1) ξ t−2n + A t A t−1 · · · A t−(2n−1) z t−2n

n = 1, 2, 3, . . .

27

Note that E[A n t ] = φ. In addition, E(A t A t−1 ) = φ 2 which can be generalized as

E(A t A t−1 A t−2 · · · A t−n ) = φ n .

autocovariances of z t are:

Therefore, provided that φ < 1, for n → ∞ the

∞

E(z t z t ) = 2σɛ 2 + σ2 η + ∑

φ i σξ 2 = 2σ2 ɛ + σ2 η + φ

(1 − φ) σ2 ξ

i=1

E(z t z t−1 ) = −σ 2 ɛ + φ2

(1 − φ) σ2 ξ

E(z t z t−2 ) =

φ 3

(1 − φ) σ2 ξ

E(z t z t−n ) = φ n−2 E(z t z t−2 ) = φn+1

(1 − φ) σ2 ξ ∀n ≥ 2 (12)

B Proof of Proposition 2

Let z t be an invertible moving average process of order two such that:

z t = a t + θ 1 a t−1 + θ 2 a t−2

Where a t is white noise process with variance σ 2 a. Considering the following autocovariance

functions: E(z 2 t ) = γ 0 , E(z t z t−1 ) = γ 1 , E(z t z t−2 ) = γ 2 . The moving

average parameters can be recovered solving the following system of three equations

(autocorrelations):

γ 2

θ 2

=

γ 0 (1 + θ1 2 + θ2 2 )

γ 1

= θ 1 + θ 1 θ 2

γ 0 (1 + θ1 2 + θ2 2 )

σa 2 γ 0

=

(1 + θ1 2 + θ2 2 )

These equations represent respectively the second and first order autocorrelations of the

process and the variance of a t .

First, the following analytical solutions can be easily obtained:

θ 2 = γ 2

σa

2 γ 1

θ 1 =

(σa 2 + γ 2 )

28

Secondly, substituting these solutions in the last equation of the system, the following

quartic equation in x (with x = σa 2 ) can be obtained:

x 4 + (2γ 2 − γ 0 )x 3 + (2γ 2 2 − 2γ 2 γ 0 + γ 2 1)x 2 + (2γ 3 2 − γ 2 2γ 0 )x + γ 4 2

x(x + γ 2 ) 2 = 0

This equation has four different solutions. Yet, the only solution leading to the

invertible process 10 is:

σa 2 = 1 (

γ 0 − 2γ 2 + G + √ √

)

2 γ0 2 + γ 0 G − 2 (γ1 2 + γ 2 (2γ 2 + G))

4

with:

and:

G = √ (γ 0 − 2γ 1 + 2γ 2 ) (γ 0 + 2γ 1 + 2γ 2 )

θ 2 =

θ 1 =

4γ

(

2

γ 0 − 2γ 2 + G + √ 2 √ )

γ0 2 + γ 0 G − 2 (γ1 2 + γ 2 (2γ 2 + G))

4γ

(

1

γ 0 + 2γ 2 + G + √ 2 √ )

γ0 2 + γ 0G − 2 (γ1 2 + γ 2 (2γ 2 + G))

10 A process with roots of the characteristic function that lie outside the unit circle (i.e. θ 2 − θ 1 <

1;−1 < θ 2 < 1).

29

TABLES and FIGURES

30

Table 1: Data description: US inventories

NAICS CODE DATA ITEM NAICS DESCRIPTION (sub-parts indicated by one or more leading dots)

31

S42 Inventories Total Merchant Wholesalers, Except Manufacturers’ Sales Branches and Offices

S423 Inventories .Durable Goods

S4231 Inventories ..Motor Vehicle & Motor Vehicle Parts & Supplies

S4232 Inventories ..Furniture & Home Furnishings

S4233 Inventories ..Lumber & Other Construction Materials

S4234 Inventories ..Professional & Commercial Equipment & Supplies

S4235 Inventories ..Metals & Minerals, Except Petroleum

S4236 Inventories ..Electrical & Electronic Goods

S4237 Inventories ..Hardware, & Plumbing & Heating Equipment & Supplies

S4238 Inventories ..Machinery, Equipment, & Supplies

S4239 Inventories ..Miscellaneous Durable Goods

S424 Inventories .Non-durable Goods

S4241 Inventories ..Paper & Paper Products

S4242 Inventories ..Drugs & Druggists’ Sundries

S4243 Inventories ..Apparel, Piece Goods, & Notions

S4244 Inventories ..Grocery & Related Products

S4245 Inventories ..Farm Product Raw Materials

S4246 Inventories ..Chemicals & Allied Products

S4247 Inventories ..Petroleum & Petroleum Products

S4248 Inventories ..Beer, Wine, & Distilled Alcoholic Beverages

S4249 Inventories ..Miscellaneous Nondurable Goods

Table 2: US inventories results – One-step-ahead forecasts – ARIMA benchmarks

MSFE ratio MSFE ratio MSFE ratio DM Test DM Test DM Test

NAICS RC-ARIMA(1,1,2) RC-ARIMA(1,1,2) RC-ARIMA(1,1,2) RC-ARIMA(1,1,2) RC-ARIMA(1,1,2) RC-ARIMA(1,1,2)

Code vs ARIMA(p,1,q) vs ARIMA(0,1,1) vs ARIMA(0,2,2) vs ARIMA(p,1,q) vs ARIMA(0,1,1) vs ARIMA(0,2,2)

32

S42 1.0852 0.6032 0.9504 0.6836 -2.6935 -1.0502

S423 0.9389 0.4849 0.9446 -0.6020 -3.0504 -0.9466

S4231 1.0157 0.9843 0.9178 0.2147 -0.2802 -1.8754

S4232 0.8665 0.6789 0.8348 -1.4714 -2.4870 -2.2998

S4233 1.0284 0.6994 0.8830 0.5261 -2.0208 -1.3062

S4234 1.0233 0.9464 0.8929 0.3010 -0.5185 -1.4342

S4236 1.0034 0.6943 0.8912 0.1081 -2.0310 -1.6263

S4237 0.9875 0.6860 0.9095 -0.3511 -2.4681 -1.7711

S4238 0.9789 0.6543 0.9467 -0.5911 -3.1448 -1.0433

S4243 0.8400 0.8754 0.8554 -2.3933 -1.9765 -2.6348

S4246 0.9353 0.8938 0.8394 -1.0814 -1.2145 -1.8469

S4248 1.0931 1.0840 1.0201 1.1137 0.9847 0.3310

EXCLUDED ITEMS

S4235 0.8517 0.6002 0.9249 -1.2227 -2.0431 -0.6058

S4239 1.0603 1.0553 0.9909 1.1566 1.1075 -0.1972

S424 0.9825 0.9973 0.9445 -0.1833 -0.0364 -0.9812

S4241 0.9708 0.9490 1.0034 -0.8955 -1.3877 0.0484

S4242 0.9895 1.0629 0.9990 -0.0600 0.8820 -0.0109

S4244 1.0083 1.0028 0.9863 0.2085 0.0701 -0.2738

S4245 0.9028 1.0472 1.0188 -1.2639 1.2075 0.4352

S4247 0.9459 1.1812 1.1658 -1.0325 1.7811 1.4787

S4249 0.9456 0.9723 0.9525 -0.7690 -0.3951 -0.5858

In-sample estimation period: 1992:01-2008:12. Out-of-sample forecasting period: 2009:01-2013:07. The benchmark

models are: ARIMA(p,1,q), ARIMA(0,1,1) and ARIMA(0,2,2). A DM test with associated p-value smaller

than 0.05 is shown in bold, implying a rejection of the null hypothesis (equal forecasting accuracy) at 5 percent

level. Negative (positive) values of the DM test indicate that the RC-ARIMA(1,1,2) performs better (worse) than

the alternative model. The excluded items are those not fulfilling conditions in Remark 3.

Table 3: US inventories results – Three-steps-ahead forecasts – ARIMA benchmarks

MSFE ratio MSFE ratio MSFE ratio DM Test DM Test DM Test

NAICS RC-ARIMA(1,1,2) RC-ARIMA(1,1,2) RC-ARIMA(1,1,2) RC-ARIMA(1,1,2) RC-ARIMA(1,1,2) RC-ARIMA(1,1,2)

Code vs ARIMA(p,1,q) vs ARIMA(0,1,1) vs ARIMA(0,2,2) vs ARIMA(p,1,q) vs ARIMA(0,1,1) vs ARIMA(0,2,2)

33

S42 1.3898 0.5021 0.8704 1.6903 -2.2595 -1.6081

S423 0.8847 0.3521 0.8426 -0.7908 -3.1390 -1.8030

S4231 0.9955 0.9852 0.8375 -0.0533 -0.1845 -2.6534

S4232 0.7705 0.5898 0.6872 -1.3159 -2.3057 -3.1987

S4233 1.0229 0.6603 0.7696 0.5001 -1.9236 -2.1726

S4234 0.9939 0.8752 0.7165 -0.0644 -0.8257 -3.1879

S4236 0.9254 0.5083 0.7135 -1.2106 -2.1356 -2.4495

S4237 0.9741 0.5105 0.8069 -0.4965 -3.0054 -2.2168

S4238 0.9501 0.5359 0.8899 -1.5079 -3.9254 -1.6519

S4243 0.8281 0.8683 0.7958 -3.1611 -2.1536 -3.7101

S4246 0.9155 0.9113 0.7569 -0.8930 -0.6862 -2.0072

S4248 1.1624 1.1330 0.9944 1.3799 1.1186 -0.0628

EXCLUDED ITEMS

S4235 0.7092 0.3938 0.7864 -2.7466 -2.2125 -1.2314

S4239 1.1098 1.0980 0.9414 1.0866 0.9894 -1.1125

S424 1.0400 1.0975 0.8649 0.4295 0.7317 -1.4318

S4241 0.9308 0.9191 0.9309 -1.6263 -1.7246 -1.2911

S4242 0.9515 1.2098 0.9567 -0.3226 1.3356 -0.3095

S4244 1.0028 1.0002 1.0733 0.0512 0.0045 0.8859

S4245 0.9014 1.0395 0.9966 -2.0853 1.0743 -0.0731

S4247 1.0315 1.0999 1.0652 0.4381 1.0973 0.5078

S4249 0.9623 0.9642 0.9044 -0.4714 -0.4526 -0.9805

In-sample estimation period: 1992:01-2008:12. Out-of-sample forecasting period: 2009:01-2013:07. The benchmark

models are: ARIMA(p,1,q), ARIMA(0,1,1) and ARIMA(0,2,2). A DM test with associated p-value smaller

than 0.05 is shown in bold, implying a rejection of the null hypothesis (equal forecasting accuracy) at 5 percent

level. Negative (positive) values of the DM test indicate that the RC-ARIMA(1,1,2) performs better (worse) than

the alternative model. The excluded items are those not fulfilling conditions in Remark 3.

Table 4: US inventories results – Six-steps-ahead forecasts – ARIMA benchmarks

MSFE ratio MSFE ratio MSFE ratio DM Test DM Test DM Test

NAICS RC-ARIMA(1,1,2) RC-ARIMA(1,1,2) RC-ARIMA(1,1,2) RC-ARIMA(1,1,2) RC-ARIMA(1,1,2) RC-ARIMA(1,1,2)

Code vs ARIMA(p,1,q) vs ARIMA(0,1,1) vs ARIMA(0,2,2) vs ARIMA(p,1,q) vs ARIMA(0,1,1) vs ARIMA(0,2,2)

34

S42 1.4934 0.5955 0.7313 2.1907 -1.7334 -2.5586

S423 0.7791 0.4279 0.6828 -1.2402 -2.7268 -2.3399

S4231 1.0471 1.0424 0.8088 0.5332 0.4820 -2.6183

S4232 0.8175 0.7254 0.5647 -0.9709 -1.3304 -4.1089

S4233 1.0645 0.7748 0.6163 1.0473 -1.2510 -2.4495

S4234 1.0809 1.0039 0.5936 0.5313 0.0199 -3.7380

S4236 0.9458 0.5960 0.5146 -0.6398 -1.6175 -3.0259

S4237 0.9934 0.6032 0.7173 -0.0956 -2.1144 -2.9908

S4238 0.9630 0.6854 0.8509 -1.2825 -2.6667 -2.0103

S4243 0.8865 0.9203 0.8102 -2.4080 -1.4419 -3.0412

S4246 1.1424 1.0714 0.7205 1.3502 0.5823 -2.1223

S4248 1.2644 1.2412 1.0543 1.8184 1.6652 0.5619

EXCLUDED ITEMS

S4235 0.5702 0.5477 0.5249 -2.5251 -1.5291 -2.3858

S4239 1.1764 1.1586 0.9406 1.9357 1.7498 -1.1693

S424 1.0636 1.1207 0.7293 0.6225 0.8615 -2.9070

S4241 0.9372 0.9215 0.9121 -0.7406 -0.8551 -1.0101

S4242 1.5525 1.5999 1.1054 2.4270 2.2479 0.6357

S4244 1.0604 1.0607 1.2226 0.9283 0.9190 1.7022

S4245 0.9609 1.0253 0.9359 -1.0868 0.8284 -1.1523

S4247 1.1440 1.1511 1.0673 2.5894 2.0848 0.4682

S4249 0.9748 0.9729 0.8389 -0.3697 -0.3965 -1.8126

In-sample estimation period: 1992:01-2008:12. Out-of-sample forecasting period: 2009:01-2013:07. The benchmark

models are: ARIMA(p,1,q), ARIMA(0,1,1) and ARIMA(0,2,2). A DM test with associated p-value smaller

than 0.05 is shown in bold, implying a rejection of the null hypothesis (equal forecasting accuracy) at 5 percent

level. Negative (positive) values of the DM test indicate that the RC-ARIMA(1,1,2) performs better (worse) than

the alternative model. The excluded items are those not fulfilling conditions in Remark 3.

Table 5: US inventories results – One-step-ahead forecasts – Exponential Smoothing (ETS) benchmarks

MSFE ratio MSFE ratio MSFE ratio DM Test DM Test DM Test

Code vs ETS(A,A d ,N) vs ETS(A,A,N) vs ETS(A,N,N) vs ETS(A,A d ,N) vs ETS(A,A,N) vs ETS(A,N,N)

35

S42 0.4995 0.9224 0.4723 -2.7188 -1.1135 -3.5305

S423 0.2805 0.8760 0.2798 -3.3297 -2.0800 -4.3943

S4231 1.0132 0.9300 1.0351 0.1811 -1.1725 0.4002

S4232 0.6506 0.8310 0.7007 -2.4817 -3.7680 -2.3724

S4233 0.5111 0.8329 0.5092 -2.8346 -2.0237 -3.3488

S4234 0.8804 0.8920 0.8362 -1.1335 -1.4516 -1.7585

S4236 0.5394 0.9109 0.5639 -2.3018 -1.0704 -2.4687

S4237 0.6001 0.8901 0.6016 -2.9022 -2.1430 -3.6374

S4238 0.5391 0.9149 0.4875 -4.0193 -1.4193 -4.4761

S4243 0.8465 0.8161 0.8497 -2.2833 -3.0451 -1.9433

S4246 0.9092 0.8510 0.8981 -1.0487 -1.9561 -1.2295

S4248 1.0443 0.9844 1.0424 0.4817 -0.4953 0.3896

EXCLUDED ITEMS

S4235 0.4403 0.4594 0.4592 -2.2651 -2.0206 -2.3807

S4239 1.0695 1.0274 1.0737 1.3302 0.6713 1.7737

S424 1.0320 0.9927 0.9690 0.2316 -0.0548 -0.2353

S4241 0.9701 0.9380 0.9814 -0.9077 -1.3333 -0.7288

S4242 0.8671 1.0296 0.8807 -1.6398 0.4174 -1.7192

S4244 1.0080 1.0211 0.9494 0.1990 0.6009 -0.9604

S4245 1.0902 1.0902 1.0902 0.9908 0.9908 1.0030

S4247 1.1889 1.1785 1.1646 1.7065 1.6106 1.5713

S4249 0.9240 0.9086 0.9329 -1.0438 -1.2825 -0.9125

models are in the Exponential Smoothing single source of error (ETS) family: The triplet (E,T,S) refers to

the three components: error, trend and seasonality; thus, for instance, the model ETS(A,A,N) has additive errors,

additive trend and no seasonality. A DM test with associated p-value smaller than 0.05 is shown in bold, implying

a rejection of the null hypothesis (equal forecasting accuracy) at 5 percent level. Negative (positive) values of the

DM test indicate that the RC-ARIMA(1,1,2) performs better (worse) than the alternative model. The excluded

items are those not fulfilling conditions in Remark 3.

Table 6: US inventories results – Three-steps-ahead forecasts – Exponential Smoothing (ETS) benchmarks

MSFE ratio MSFE ratio MSFE ratio DM Test DM Test DM Test

Code vs ETS(A,A d ,N) vs ETS(A,A,N) vs ETS(A,N,N) vs ETS(A,A d ,N) vs ETS(A,A,N) vs ETS(A,N,N)

36

S42 0.4202 0.8243 0.3873 -2.6093 -2.4417 -3.7653

S423 0.2857 0.8244 0.2802 -3.2985 -2.2483 -4.7399

S4231 0.9922 0.9066 1.0120 -0.0926 -1.1682 0.1227

S4232 0.5688 0.7145 0.6256 -2.4065 -4.3704 -2.4678

S4233 0.5915 0.7559 0.5807 -2.1934 -2.4959 -2.8512

S4234 0.8391 0.7337 0.7512 -1.0393 -2.6709 -2.2285

S4236 0.4368 0.6965 0.4588 -2.2861 -2.9189 -2.6846

S4237 0.4693 0.7782 0.4522 -3.2178 -2.7040 -5.0564

S4238 0.4811 0.8775 0.4263 -4.3408 -1.8889 -4.9614

S4243 0.8531 0.8239 0.8583 -2.2952 -2.7321 -2.0658

S4246 0.9168 0.7424 0.8745 -0.6573 -2.2955 -1.0721

S4248 1.2033 0.9588 1.0225 1.6199 -1.0128 0.1528

EXCLUDED ITEMS

S4235 0.3228 0.4063 0.3496 -2.5445 -1.9361 -2.6145

S4239 1.1119 1.0454 1.1125 1.0876 0.5610 1.4515

S424 1.0479 0.9045 0.9127 0.3087 -0.8431 -0.5849

S4241 0.9294 0.9126 0.9801 -1.6143 -1.7437 -1.3077

S4242 1.0940 1.0669 0.9973 0.6860 0.5389 -0.0826

S4244 1.0009 1.0305 0.8478 0.0161 0.6563 -2.2798

S4245 1.0330 1.0330 1.0330 0.4887 0.4887 0.5057

S4247 1.1056 1.0787 1.0542 1.0726 0.7879 0.6613

S4249 0.9317 0.9092 0.9363 -0.8234 -1.1593 -0.6876

models are in the Exponential Smoothing single source of error (ETS) family: The triplet (E,T,S) refers to

the three components: error, trend and seasonality; thus, for instance, the model ETS(A,A,N) has additive errors,

additive trend and no seasonality. A DM test with associated p-value smaller than 0.05 is shown in bold, implying

a rejection of the null hypothesis (equal forecasting accuracy) at 5 percent level. Negative (positive) values of the

DM test indicate that the RC-ARIMA(1,1,2) performs better (worse) than the alternative model. The excluded

items are those not fulfilling conditions in Remark 3.

Table 7: US inventories results – Six-steps-ahead forecasts – Exponential Smoothing (ETS) benchmarks

MSFE ratio MSFE ratio MSFE ratio DM Test DM Test DM Test

Code vs ETS(A,A d ,N) vs ETS(A,A,N) vs ETS(A,N,N) vs ETS(A,A d ,N) vs ETS(A,A,N) vs ETS(A,N,N)

37

S42 0.5312 0.7206 0.4617 -2.0250 -3.0398 -3.3875

S423 0.3779 0.6942 0.3517 -2.9014 -2.3537 -4.7372

S4231 1.0422 0.9021 1.0324 0.4717 -1.0817 0.2845

S4232 0.7086 0.6029 0.7765 -1.4136 -4.5974 -1.3393

S4233 0.7283 0.6407 0.6765 -1.4544 -2.4428 -2.4699

S4234 0.9830 0.6214 0.7912 -0.0851 -2.9978 -1.5750

S4236 0.5413 0.4975 0.5520 -1.7621 -3.6257 -2.3316

S4237 0.5716 0.7067 0.5342 -2.2811 -3.3832 -4.1587

S4238 0.6489 0.8560 0.5677 -2.9529 -2.0332 -3.5481

S4243 0.9105 0.8738 0.9165 -1.6142 -2.0034 -1.7403

S4246 1.0715 0.7332 0.9080 0.5910 -1.9753 -0.8913

S4248 1.2970 0.9296 1.0147 1.9880 -1.2973 0.0847

EXCLUDED ITEMS

S4235 0.4355 0.3813 0.5205 -2.1297 -2.6863 -1.7948

S4239 1.1756 1.1327 1.1078 1.8892 1.5645 1.5754

S424 1.0990 0.7942 0.8997 0.6369 -1.9352 -0.6543

S4241 0.9295 0.9054 0.9932 -0.8091 -1.0014 -0.3175

S4242 1.5683 1.2618 1.0108 2.3374 1.6601 0.3830

S4244 1.0571 1.0766 0.8085 0.8703 1.2577 -3.3891

S4245 1.0380 1.0380 1.0477 0.8040 0.8040 1.1156

S4247 1.1649 1.1257 1.0434 2.1301 1.5694 0.7967

S4249 0.9545 0.9260 0.9895 -0.6656 -1.2294 -0.1325

models are in the Exponential Smoothing single source of error (ETS) family: The triplet (E,T,S) refers to

the three components: error, trend and seasonality; thus, for instance, the model ETS(A,A,N) has additive errors,

additive trend and no seasonality. A DM test with associated p-value smaller than 0.05 is shown in bold, implying

a rejection of the null hypothesis (equal forecasting accuracy) at 5 percent level. Negative (positive) values of the

DM test indicate that the RC-ARIMA(1,1,2) performs better (worse) than the alternative model. The excluded

items are those not fulfilling conditions in Remark 3.

Figure B.1: Industrial Production Index and Merchant Wholesalers Inventories for the

US

120

INDPRO (2007=100) INVENT (2007=100)

110

100

90

80

70

60

50

1993 1995 1997 1999 2001 2003 2005 2007 2009 2011 2013

This graph shows the Industrial Production Index (INDPRO), Index 2007=100, Monthly, Seasonally

Adjusted; Merchant Wholesalers Inventories (WHLSLRIMSA), Index 2007=100, Monthly, Seasonally

Adjusted. Sample: 1992:01-2013:09. Source: Federal Reserve Bank of St. Louis.

38

Figure B.2: Simulated time series with switching dynamic behavior

20 40 60 80 100 120 140 160

The above chart shows a simulated model as in (1) and (2). In the shaded areas the series behaves as a

simple exponential smoothing dynamics, while in the non-shaded area the series behaves as a local linear

trend.

39

Figure B.3: Variance (σ 2 a ) 0.6

0.0

0.2

0.4

1.0

0.8

0.6

0.4

0.4

0.2

0.0

0.2

The above chart shows the variance (vertical axis) as a function of γ 1 and γ 2 having fixed γ 0 = 1.

Figure B.4: Variance (σ 2 a)

0.5

0.4

0.3

0.2

0.1

0.0

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

The above chart shows the contour plot of the variance by changing γ 1 and γ 2 .

40

Figure B.5: θ 1

0.0

0.5

1.0

0.4

0.2

0.0

0.2

0.4

0.6

0.0

The above chart shows θ 1 as a function of γ 1 and γ 2 , having fixed γ 0 = 1.

Figure B.6: θ 1

0.5

0.4

0.3

0.2

0.1

0.0

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

The above chart shows the contour plot of θ 1 as a function of γ 1 and γ 2 , having fixed γ 0 = 1.

41

Figure B.7: θ 2

0.6

0.4

0.2

0.0

1.0

0.5

0.0

0.2

0.4

0.0

The above chart shows θ 2 as a function of γ 1 and γ 2 , having fixed γ 0 = 1.

Figure B.8: θ 2

0.5

0.4

0.3

0.2

0.1

0.0

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0

The above chart shows the contour plot of θ 2 as a function of γ 1 and γ 2 , having fixed γ 0 = 1.

42

Figure B.9: US inventories: time series plots

S423

S4231

S4232

S4233

S4234

350

60

10

16

40

300

250

50

40

9

8

7

14

12

10

35

30

200

150

30

6

5

8

6

25

20

100

1995 2000 2005 2010

20

1995 2000 2005 2010

4

1995 2000 2005 2010

4

1995 2000 2005 2010

15

1995 2000 2005 2010

S4235

S4236

S4237

S4238

S4239

32

40

20

100

30

28

24

35

30

16

80

25

20

12

60

20

16

12

25

20

8

40

15

43

8

1995 2000 2005 2010

15

1995 2000 2005 2010

4

1995 2000 2005 2010

20

1995 2000 2005 2010

10

1995 2000 2005 2010

S424

S4241

S4242

S4243

S4244

200

8

40

24

35

160

7

30

20

30

120

6

20

16

25

80

5

10

12

20

40

1995 2000 2005 2010

4

1995 2000 2005 2010

0

1995 2000 2005 2010

8

1995 2000 2005 2010

15

1995 2000 2005 2010

S4245

S4246

S4247

S4248

S4249

35

14

30

16

28

30

12

25

14

24

25

10

20

12

20

20

8

15

10

15

6

10

8

16

10

4

5

6

12

5

1995 2000 2005 2010

2

1995 2000 2005 2010

0

1995 2000 2005 2010

4

1995 2000 2005 2010

8

1995 2000 2005 2010

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Financial Crisis: Issues in Business, Finance and Global Economics, New York, Nova Science

Publishers, Inc., TD No. 749 (March 2010).

A. LEVY and A. ZAGHINI, The pricing of government guaranteed bank bonds, Banks and Bank Systems, v.

6, 3, pp. 16-24, TD No. 753 (March 2010).

G. BARONE, R. FELICI and M. PAGNINI, Switching costs in local credit markets, International Journal of

Industrial Organization, v. 29, 6, pp. 694-704, TD No. 760 (June 2010).

G. BARBIERI, C. ROSSETTI e P. SESTITO, The determinants of teacher mobility: evidence using Italian

teachers' transfer applications, Economics of Education Review, v. 30, 6, pp. 1430-1444,

TD No. 761 (marzo 2010).

G. GRANDE and I. VISCO, A public guarantee of a minimum return to defined contribution pension scheme

members, The Journal of Risk, v. 13, 3, pp. 3-43, TD No. 762 (June 2010).

P. DEL GIOVANE, G. ERAMO and A. NOBILI, Disentangling demand and supply in credit developments: a

survey-based analysis for Italy, Journal of Banking and Finance, v. 35, 10, pp. 2719-2732, TD No.

764 (June 2010).

G. BARONE and S. MOCETTI, With a little help from abroad: the effect of low-skilled immigration on the

female labour supply, Labour Economics, v. 18, 5, pp. 664-675, TD No. 766 (July 2010).

S. FEDERICO and A. FELETTIGH, Measuring the price elasticity of import demand in the destination markets of

italian exports, Economia e Politica Industriale, v. 38, 1, pp. 127-162, TD No. 776 (October 2010).

S. MAGRI and R. PICO, The rise of risk-based pricing of mortgage interest rates in Italy, Journal of

Banking and Finance, v. 35, 5, pp. 1277-1290, TD No. 778 (October 2010).

M. TABOGA, Under/over-valuation of the stock market and cyclically adjusted earnings, International

Finance, v. 14, 1, pp. 135-164, TD No. 780 (December 2010).

S. NERI, Housing, consumption and monetary policy: how different are the U.S. and the Euro area, Journal

of Banking and Finance, v.35, 11, pp. 3019-3041, TD No. 807 (April 2011).

V. CUCINIELLO, The welfare effect of foreign monetary conservatism with non-atomistic wage setters, Journal

of Money, Credit and Banking, v. 43, 8, pp. 1719-1734, TD No. 810 (June 2011).

A. CALZA and A. ZAGHINI, welfare costs of inflation and the circulation of US currency abroad, The B.E.

Journal of Macroeconomics, v. 11, 1, Art. 12, TD No. 812 (June 2011).

I. FAIELLA, La spesa energetica delle famiglie italiane, Energia, v. 32, 4, pp. 40-46, TD No. 822 (September

2011).

D. DEPALO and R. GIORDANO, The public-private pay gap: a robust quantile approach, Giornale degli

Economisti e Annali di Economia, v. 70, 1, pp. 25-64, TD No. 824 (September 2011).

R. DE BONIS and A. SILVESTRINI, The effects of financial and real wealth on consumption: new evidence from

OECD countries, Applied Financial Economics, v. 21, 5, pp. 409–425, TD No. 837 (November 2011).

F. CAPRIOLI, P. RIZZA and P. TOMMASINO, Optimal fiscal policy when agents fear government default, Revue

Economique, v. 62, 6, pp. 1031-1043, TD No. 859 (March 2012).

2012

F. CINGANO and A. ROSOLIA, People I know: job search and social networks, Journal of Labor Economics, v.

30, 2, pp. 291-332, TD No. 600 (September 2006).

G. GOBBI and R. ZIZZA, Does the underground economy hold back financial deepening Evidence from the

italian credit market, Economia Marche, Review of Regional Studies, v. 31, 1, pp. 1-29, TD No. 646

(November 2006).

S. MOCETTI, Educational choices and the selection process before and after compulsory school, Education

Economics, v. 20, 2, pp. 189-209, TD No. 691 (September 2008).

P. PINOTTI, M. BIANCHI and P. BUONANNO, Do immigrants cause crime, Journal of the European

Economic Association , v. 10, 6, pp. 1318–1347, TD No. 698 (December 2008).

M. PERICOLI and M. TABOGA, Bond risk premia, macroeconomic fundamentals and the exchange rate,

International Review of Economics and Finance, v. 22, 1, pp. 42-65, TD No. 699 (January 2009).

F. LIPPI and A. NOBILI, Oil and the macroeconomy: a quantitative structural analysis, Journal of European

Economic Association, v. 10, 5, pp. 1059-1083, TD No. 704 (March 2009).

G. ASCARI and T. ROPELE, Disinflation in a DSGE perspective: sacrifice ratio or welfare gain ratio,

Journal of Economic Dynamics and Control, v. 36, 2, pp. 169-182, TD No. 736 (January 2010).

S. FEDERICO, Headquarter intensity and the choice between outsourcing versus integration at home or

abroad, Industrial and Corporate Chang, v. 21, 6, pp. 1337-1358, TD No. 742 (February 2010).

I. BUONO and G. LALANNE, The effect of the Uruguay Round on the intensive and extensive margins of

trade, Journal of International Economics, v. 86, 2, pp. 269-283, TD No. 743 (February 2010).

A. BRANDOLINI, S. MAGRI and T. M SMEEDING, Asset-based measurement of poverty, In D. J. Besharov

and K. A. Couch (eds), Counting the Poor: New Thinking About European Poverty Measures and

Lessons for the United States, Oxford and New York: Oxford University Press, TD No. 755

(March 2010).

S. GOMES, P. JACQUINOT and M. PISANI, The EAGLE. A model for policy analysis of macroeconomic

interdependence in the euro area, Economic Modelling, v. 29, 5, pp. 1686-1714, TD No. 770

(July 2010).

A. ACCETTURO and G. DE BLASIO, Policies for local development: an evaluation of Italy’s “Patti

Territoriali”, Regional Science and Urban Economics, v. 42, 1-2, pp. 15-26, TD No. 789

(January 2006).

F. BUSETTI and S. DI SANZO, Bootstrap LR tests of stationarity, common trends and cointegration, Journal

of Statistical Computation and Simulation, v. 82, 9, pp. 1343-1355, TD No. 799 (March 2006).

S. NERI and T. ROPELE, Imperfect information, real-time data and monetary policy in the Euro area, The

Economic Journal, v. 122, 561, pp. 651-674, TD No. 802 (March 2011).

A. ANZUINI and F. FORNARI, Macroeconomic determinants of carry trade activity, Review of International

Economics, v. 20, 3, pp. 468-488, TD No. 817 (September 2011).

M. AFFINITO, Do interbank customer relationships exist And how did they function in the crisis Learning

from Italy, Journal of Banking and Finance, v. 36, 12, pp. 3163-3184, TD No. 826 (October 2011).

P. GUERRIERI and F. VERGARA CAFFARELLI, Trade Openness and International Fragmentation of

Production in the European Union: The New Divide, Review of International Economics, v. 20, 3,

pp. 535-551, TD No. 855 (February 2012).

V. DI GIACINTO, G. MICUCCI and P. MONTANARO, Network effects of public transposrt infrastructure:

evidence on Italian regions, Papers in Regional Science, v. 91, 3, pp. 515-541, TD No. 869 (July

2012).

A. FILIPPIN and M. PACCAGNELLA, Family background, self-confidence and economic outcomes,

Economics of Education Review, v. 31, 5, pp. 824-834, TD No. 875 (July 2012).

2013

A. MERCATANTI, A likelihood-based analysis for relaxing the exclusion restriction in randomized

experiments with imperfect compliance, Australian and New Zealand Journal of Statistics, v. 55, 2,

pp. 129-153, TD No. 683 (August 2008).

F. CINGANO and P. PINOTTI, Politicians at work. The private returns and social costs of political connections,

Journal of the European Economic Association, v. 11, 2, pp. 433-465, TD No. 709 (May 2009).

F. BUSETTI and J. MARCUCCI, Comparing forecast accuracy: a Monte Carlo investigation, International

Journal of Forecasting, v. 29, 1, pp. 13-27, TD No. 723 (September 2009).

D. DOTTORI, S. I-LING and F. ESTEVAN, Reshaping the schooling system: The role of immigration, Journal

of Economic Theory, v. 148, 5, pp. 2124-2149, TD No. 726 (October 2009).

A. FINICELLI, P. PAGANO and M. SBRACIA, Ricardian Selection, Journal of International Economics, v. 89,

1, pp. 96-109, TD No. 728 (October 2009).

L. MONTEFORTE and G. MORETTI, Real-time forecasts of inflation: the role of financial variables, Journal

of Forecasting, v. 32, 1, pp. 51-61, TD No. 767 (July 2010).

R. GIORDANO and P. TOMMASINO, Public-sector efficiency and political culture, FinanzArchiv, v. 69, 3, pp.

289-316, TD No. 786 (January 2011).

E. GAIOTTI, Credit availablility and investment: lessons from the "Great Recession", European Economic

Review, v. 59, pp. 212-227, TD No. 793 (February 2011).

F. NUCCI and M. RIGGI, Performance pay and changes in U.S. labor market dynamics, Journal of

Economic Dynamics and Control, v. 37, 12, pp. 2796-2813, TD No. 800 (March 2011).

G. CAPPELLETTI, G. GUAZZAROTTI and P. TOMMASINO, What determines annuity demand at retirement,

The Geneva Papers on Risk and Insurance – Issues and Practice, pp. 1-26, TD No. 805 (April 2011).

A. ACCETTURO e L. INFANTE, Skills or Culture An analysis of the decision to work by immigrant women

in Italy, IZA Journal of Migration, v. 2, 2, pp. 1-21, TD No. 815 (July 2011).

A. DE SOCIO, Squeezing liquidity in a “lemons market” or asking liquidity “on tap”, Journal of Banking and

Finance, v. 27, 5, pp. 1340-1358, TD No. 819 (September 2011).

S. GOMES, P. JACQUINOT, M. MOHR and M. PISANI, Structural reforms and macroeconomic performance

in the euro area countries: a model-based assessment, International Finance, v. 16, 1, pp. 23-44,

TD No. 830 (October 2011).

G. BARONE and G. DE BLASIO, Electoral rules and voter turnout, International Review of Law and

Economics, v. 36, 1, pp. 25-35, TD No. 833 (November 2011).

O. BLANCHARD and M. RIGGI, Why are the 2000s so different from the 1970s A structural interpretation

of changes in the macroeconomic effects of oil prices, Journal of the European Economic

Association, v. 11, 5, pp. 1032-1052, TD No. 835 (November 2011).

R. CRISTADORO and D. MARCONI, Household savings in China, in G. Gomel, D. Marconi, I. Musu, B.

Quintieri (eds), The Chinese Economy: Recent Trends and Policy Issues, Springer-Verlag, Berlin,

TD No. 838 (November 2011).

E. GENNARI and G. MESSINA, How sticky are local expenditures in Italy Assessing the relevance of the

flypaper effect through municipal data, International Tax and Public Finance (DOI:

10.1007/s10797-013-9269-9), TD No. 844 (January 2012).

A. ANZUINI, M. J. LOMBARDI and P. PAGANO, The impact of monetary policy shocks on commodity prices,

International Journal of Central Banking, v. 9, 3, pp. 119-144, TD No. 851 (February 2012).

R. GAMBACORTA and M. IANNARIO, Measuring job satisfaction with CUB models, Labour, v. 27, 2, pp.

198-224, TD No. 852 (February 2012).

G. ASCARI and T. ROPELE, Disinflation effects in a medium-scale new keynesian model: money supply rule

versus interest rate rule, European Economic Review, v. 61, pp. 77-100, TD No. 867 (April

2012).

E. BERETTA and S. DEL PRETE, Banking consolidation and bank-firm credit relationships: the role of

geographical features and relationship characteristics, Review of Economics and Institutions,

v. 4, 3, pp. 1-46, TD No. 901 (February 2013).

M. ANDINI, G. DE BLASIO, G. DURANTON and W. STRANGE, Marshallian labor market pooling: evidence

from Italy, Regional Science and Urban Economics, v. 43, 6, pp.1008-1022, TD No. 922 (July

2013).

G. SBRANA and A. SILVESTRINI, Forecasting aggregate demand: analytical comparison of top-down and

bottom-up approaches in a multivariate exponential smoothing framework, International Journal of

Production Economics, v. 146, 1, pp. 185-98, TD No. 929 (September 2013).

A. FILIPPIN, C. V, FIORIO and E. VIVIANO, The effect of tax enforcement on tax morale, European Journal

of Political Economy, v. 32, pp. 320-331, TD No. 937 (October 2013).

2014

M. TABOGA, The riskiness of corporate bonds, Journal of Money, Credit and Banking, v.46, 4, pp. 693-713,

TD No. 730 (October 2009).

G. MICUCCI and P. ROSSI, Il ruolo delle tecnologie di prestito nella ristrutturazione dei debiti delle imprese in

crisi, in A. Zazzaro (a cura di), Le banche e il credito alle imprese durante la crisi, Bologna, Il Mulino,

TD No. 763 (June 2010).

P. ANGELINI, S. NERI and F. PANETTA, The interaction between capital requirements and monetary policy,

Journal of Money, Credit and Banking, v. 46, 6, pp. 1073-1112, TD No. 801 (March 2011).

M. FRANCESE and R. MARZIA, Is there Room for containing healthcare costs An analysis of regional

spending differentials in Italy, The European Journal of Health Economics, v. 15, 2, pp. 117-132,

TD No. 828 (October 2011).

L. GAMBACORTA and P. E. MISTRULLI, Bank heterogeneity and interest rate setting: what lessons have we

learned since Lehman Brothers, Journal of Money, Credit and Banking, v. 46, 4, pp. 753-778,

TD No. 829 (October 2011).

M. PERICOLI, Real term structure and inflation compensation in the euro area, International Journal of

Central Banking, v. 10, 1, pp. 1-42, TD No. 841 (January 2012).

V. DI GACINTO, M. GOMELLINI, G. MICUCCI and M. PAGNINI, Mapping local productivity advantages in Italy:

industrial districts, cities or both, Journal of Economic Geography, v. 14, pp. 365–394, TD No. 850

(January 2012).

A. ACCETTURO, F. MANARESI, S. MOCETTI and E. OLIVIERI, Don't Stand so close to me: the urban impact

of immigration, Regional Science and Urban Economics, v. 45, pp. 45-56, TD No. 866 (April

2012).

S. FEDERICO, Industry dynamics and competition from low-wage countries: evidence on Italy, Oxford

Bulletin of Economics and Statistics, v. 76, 3, pp. 389-410, TD No. 879 (September 2012).

F. D’AMURI and G. PERI, Immigration, jobs and employment protection: evidence from Europe before and

during the Great Recession, Journal of the European Economic Association, v. 12, 2, pp. 432-464,

TD No. 886 (October 2012).

M. TABOGA, What is a prime bank A euribor-OIS spread perspective, International Finance, v. 17, 1, pp.

51-75, TD No. 895 (January 2013).

L. GAMBACORTA and F. M. SIGNORETTI, Should monetary policy lean against the wind An analysis based

on a DSGE model with banking, Journal of Economic Dynamics and Control, v. 43, pp. 146-74,

TD No. 921 (July 2013).

U. ALBERTAZZI and M. BOTTERO, Foreign bank lending: evidence from the global financial crisis, Journal

of International Economics, v. 92, 1, pp. 22-35, TD No. 926 (July 2013).

R. DE BONIS and A. SILVESTRINI, The Italian financial cycle: 1861-2011, Cliometrica, v.8, 3, pp. 301-334,

TD No. 936 (October 2013).

D. PIANESELLI and A. ZAGHINI, The cost of firms’ debt financing and the global financial crisis, Finance

Research Letters, v. 11, 2, pp. 74-83, TD No. 950 (February 2014).

A. ZAGHINI, Bank bonds: size, systemic relevance and the sovereign, International Finance, v. 17, 2, pp. 161-

183, TD No. 966 (July 2014).

FORTHCOMING

M. BUGAMELLI, S. FABIANI and E. SETTE, The age of the dragon: the effect of imports from China on firmlevel

prices, Journal of Money, Credit and Banking, TD No. 737 (January 2010).

F. D’AMURI, Gli effetti della legge 133/2008 sulle assenze per malattia nel settore pubblico, Rivista di

Politica Economica, TD No. 787 (January 2011).

E. COCOZZA and P. PISELLI, Testing for east-west contagion in the European banking sector during the

financial crisis, in R. Matoušek; D. Stavárek (eds.), Financial Integration in the European Union,

Taylor & Francis, TD No. 790 (February 2011).

R. BRONZINI and E. IACHINI, Are incentives for R&D effective Evidence from a regression discontinuity

approach, American Economic Journal : Economic Policy, TD No. 791 (February 2011).

G. DE BLASIO, D. FANTINO and G. PELLEGRINI, Evaluating the impact of innovation incentives: evidence

from an unexpected shortage of funds, Industrial and Corporate Change, TD No. 792 (February

2011).

A. DI CESARE, A. P. STORK and C. DE VRIES, Risk measures for autocorrelated hedge fund returns, Journal

of Financial Econometrics, TD No. 831 (October 2011).

D. FANTINO, A. MORI and D. SCALISE, Collaboration between firms and universities in Italy: the role of a

firm's proximity to top-rated departments, Rivista Italiana degli economisti, TD No. 884 (October

2012).

G. BARONE and S. MOCETTI, Natural disasters, growth and institutions: a tale of two earthquakes, Journal

of Urban Economics, TD No. 949 (January 2014).